Quantum Harmonic Oscillator

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At sufficiently small energies, the harmonic oscillator as governed by the laws of quantum mechanics, known simply as the quantum harmonic oscillator, differs significantly from its description according to the laws of classical physics. Whereas the energy of the classical harmonic oscillator is allowed to take on any positive value, the quantum harmonic oscillator has discrete energy levels \( E_n \) given by

\[ E_n = \hbar \omega \left(n + \frac{1}{2}\right), \]

where \( \hbar \) is the Planck constant, \( \omega \) is the (classical) angular frequency, and \( n \) is a non-negative integer. Furthermore, whereas the classical harmonic oscillator is confined to a finite region of space, the quantum harmonic oscillator has a nonzero (but asymptotically vanishing) probability of being found anywhere.

As one of the few important quantum mechanical systems whose dynamics can be determined exactly, the quantum harmonic oscillator frequently serves as a basis for describing many real-world phenomena, such as molecular vibrations.

where \( \hat{x} \) and \( \hat{p} \) are the position and momentum operators, respectively. The ladder operator method to obtain the eigenfunctions and energies (due to Dirac) is to factor the equation by using the so-called lowering operator

where \( H_n \) is the \( n \)th Hermite polynomial. A plot of the first few wavefunctions is shown below.

Since the harmonic potential is non-vanishing, all of the eigenstates are bound and the energy spectrum discrete, although as is characteristic in quantum mechanics, the wavefunctions extend to all space (zero only at nodes). This should be contrasted with the classical harmonic oscillator, whose probability density is bounded by the amplitude of its oscillation and whose energies are continuous. Below is the probability density of the ground state of the quantum harmonic oscillator compared with the U-shaped density of the classical oscillator.

Harmonic oscillator ladder operators

Using the ladder operators, many dynamical quantities can be calculated for the harmonic oscillator without direct integration. One can express the position and momentum operators as follows:

Show that the ground state is the only eigenstate of the harmonic oscillator that is a minimum-uncertainty state in position-momentum space (i.e., equality holds for the Heisenberg uncertainty relation in \( x \) and \( p \)).

which achieves the mininmum uncertainty of \( \hbar/2 \) only for the ground state \( n = 0. \)

Applications

Many practical potentials can be treated (or at least closely approximated) as harmonic potentials. The internuclear potential well of a molecule of diatomic gas, such as molecular oxygen or nitrogen, can be taken as a harmonic potential, in which case the vibrational energy levels \( \epsilon_n \) are given by

\[ \epsilon_n = \hbar \omega \left( n + \frac{1}{2} \right), \]

where there vibrational angular frequency \( \omega \) can be computed from the force constant of the molecule and its reduced mass.